Claims

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1967_06_01_hooley: Hooley (Mathematika 14, 1967) proved that the n up to x at which an irreducible integer polynomial of degree k at least 3 takes (k-1)-power-free values number c x + o(x), with c the Euler product of the local factors.

2006_01_01_heath_brown: Theorem 16 of Heath-Brown's 2006 lecture notes proves the Euler-product asymptotic for the r-free values of an irreducible integer polynomial of degree d whenever r is at least (3d+2)/4, reaching r = d-2 for d at least 10.

2011_02_19_browning: Browning (Arch. Math. 96, 2011) proves the Euler-product asymptotic for the r-free values of an irreducible integer polynomial of degree d whenever r is at least (3d+1)/4, which reaches r = d-2 for d at least 9; refereed.

2023_10_25_carella: Carella's 2023 preprint claims asymptotic counts of the squarefree values of n^4+1, n^4+2 and n^4-2n^2+2, which would answer the question about n^4+2; unrefereed and unreviewed.

2026_08_11_zapata_ceballos_jalalvand: Zapata Ceballos and Jalalvand's 2026 preprint claims a positive density of squarefree values for monic irreducible degree-2q polynomials whose field has a Galois subfield of degree q, which includes n^4+2; unreviewed.

2026_09_24_openai: OpenAI's release proves that an irreducible integer polynomial of degree k at least 4 with no fixed prime (k-2)-th power divisor takes (k-2)-power-free values on a positive-density set, so n^4 + 2 is squarefree infinitely often.

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